I am about to enter a PhD program, and I am very interested in learning about opportunities that allow me to go abroad after I finish. During my undergraduate career, I studied in China for a year, and I really enjoyed the environment of learning new languages and living in a different culture.
Specifically, I want to know about some of the challenges that one might have in looking for university or research positions in non-Western countries, and subsequently, how one might transition back to the United States after having spent a considerable amount of time abroad (5+ years). I have felt that math is a universal topic, and many universities abroad (especially in Asian and central Asian countries) would have faculty positions for English speaking PhDs to move there. However, I don't have any information to justify these beliefs.
I'm not as concerned about salary or ending up at a prestigious research university. I really enjoy teaching and working with students, and I enjoy math enough that any position that would allow me to continue working in mathematics and with students would be acceptable. Am I entirely off base in hoping to find positions abroad? What are some challenges that I might not have considered?
EDIT: I should have clarified. I am an American student beginning a program in the United States. In that context, then, what are things that an American PhD should be aware of when looking for opportunities in non-Western countries, such as those in Southeast Asia, Central Asia, Africa, etc.?